It is possible, but requires a lot of configuration work and the user experience will be mediocre.
Main problem is that Speech-to-Text (STT) and Text-to-Speech (TTS) require a lot of computing power - that is the reason why external services are used.

You can run STT locally e.g. with PocketSphinx - the voice recognition will be poor and you will have a lot of “misunderstandings” or your voice will not be understood by Mycroft at all.
For TTS you can use for example espeak, but the voice will sound very robotic/unatural.